Answer
$u_x=\frac{yx^{\frac{y}{z}-1}}{z}$, $f_y=\frac{ln{(x)}x^{\frac{y}{z}}}{z}$, $f_z=\frac{-yln{(x)}x^{\frac{y}{z}}}{z^2}$.
Work Step by Step
$u=x^{\frac{y}{z}}$
In order to find $u_x$ we treat $y$ and $z$ as constants and differentiate with respect to $x$.
$u_x=\frac{yx^{\frac{y}{z}-1}}{z}$
Analogously:
$f_y=\frac{ln{(x)}x^{\frac{y}{z}}}{z}$
$f_z=\frac{-yln{(x)}x^{\frac{y}{z}}}{z^2}$
